A Universe with too much matter-and-energy for its expansion rate will recollapse in short order; a Universe with too little will expand into oblivion before it’s possible to even form atoms. Yet not only has our Universe neither recollapsed nor failed to yield atoms, but even today, some 13.8 billion years after the Big Bang, those two sides of the equation appear to be perfectly in balance.
If we extrapolate this back to a very early time — say, one nanosecond after the hot Big Bang — we find that not only do these two sides have to balance, but they have to balance to an extraordinary precision. The Universe’s initial expansion rate and the sum total of all the different forms of matter and energy in the Universe not only need to balance, but they need to balance to more than 20 significant digits. It’s like guessing the same 1-to-1,000,000 number as me three times in a row, and then predicting the outcome of 16 consecutive coin-flips immediately afterwards.
The odds of this occurring naturally, if we consider all the random possibilities we could have imagined, are astronomically small.